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MarginalApproach - Chapter 12 Population-averaged models...

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Chapter 12: Population-averaged models for Bernoulli repeated measurements Example of repeated measures: Data are comprised of several repeated measurements on the same individual over time, e.g. Y ij = 1 indicates acne outbreak for patient i in month j ; Y ij = 0 indicates no outbreak. Data are recorded in clusters, e.g. Y ij might indicate the presence of tooth decay for tooth j in patient i . Data are from naturally associated groups, e.g. Y ij might denote a successful treatment of patient j at clinic i . Wheezing data in Chapter 1. In all of these examples, the repeated measurements are (typically positively) correlated within an individual or group. 1
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Marginal logistic model of multiple 0/1 responses Let n i binary responses Y i = ( Y i 1 , . . . , Y in i ) come from the i th individual at times t i = ( t i 1 , . . . , t in i ). Let π i = ( π i 1 , . . . , π in i ) where π ij = E ( Y ij ). Let x ij be a p × 1 vector of explanatory variables. We assume the vectors Y 1 , . . . , Y n are independent, but that elements of Y i are correlated. Common choices are R ( α ) = corr( Y i ) = 1 α α · · · α α 1 α · · · α α α 1 · · · α . . . . . . . . . . . . . . . α α α · · · 1 n i × n i exchangeable, and R ( α ) = corr( Y i ) = 1 α α 2 · · · α n i - 1 α 1 α · · · α n i - 2 α 2 α 1 · · · α n i - 3 . . . . . . . . . . . . . . . α n i - 1 α n i - 2 α n i - 3 · · · 1 n i × n i AR(1). 2
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Others are R ( α ) = corr( Y i ) = 1 α 12 α 13 · · · α 1 n α 12 1 α 23 · · · α 2 n α 13 α 23 1 · · · α 3 n . . . . . . . . . . . . . . . α 1 n α 2 n α 3 n · · · 1 n × n unstructured, and R = corr( Y i ) = 1 0 0 · · · 0 0 1 0 · · · 0 0 0 1 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · 1 n i × n i independence. 3
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You can also specify a fixed, known R as well as MDEP( m ) which yields R ( α ) as corr( Y ij , Y i,j + t ) = 1 t = 0 α t t = 1 , . . . , m 0 t > m . Unstructured most general; often a default choice. Need balance though. Exchangeable useful when time is not important and correlations thought to be approximately equal, e.g. repeated measurements on individual in crossover study, measurements across several individuals from clinic i . AR(1) useful when serial correlation plausible, e.g. repeated measurements across equally spaced time points on individual. 4
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Comments: These correlation matrices are used in a GEE algorithm (sketched below) in PROC GENMOD (all other PROC MIXED covariance structures available in GLIMMIX). Repeated measures are accounted for via REPEATED statement. The order of ( Y i 1 , . . . , Y in ) makes a difference with some R ( α ). If ordering is different to that defined in the DATA step, one can use the WITHIN subcommand in the REPEATED statement to tell SAS what the ordering is. Also used when missing some measurements in ( Y i 1 , . . . , Y in ).
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